Mathematics > Algebraic Topology

Title:
Spectra and symmetric spectra in general model categories

Abstract: (This is an updated version; following an idea of Voevodsky, we have
strengthened our results so all of them apply to one form of motivic homotopy
theory).
We give two general constructions for the passage from unstable to stable
homotopy that apply to the known example of topological spaces, but also to new
situations, such as motivic homotopy theory of schemes. One is based on the
standard notion of spectra originated by Boardman. Its input is a well-behaved
model category C and an endofunctor G, generalizing the suspension. Its output
is a model category on which G is a Quillen equivalence. Under strong
hypotheses the weak equivalences in this model structure are the appropriate
analogue of stable homotopy isomorphisms.
The second construction is based on symmetric spectra, and is of value only
when C has some monoidal structure that G preserves. In this case, ordinary
spectra generally will not have monoidal structure, but symmetric spectra will.
Our abstract approach makes constructing the stable model category of symmetric
spectra straightforward. We study properties of these stabilizations; most
importantly, we show that the two different stabilizations are Quillen
equivalent under some hypotheses (that also hold in the motivic example).